1. NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 12: The beautiful theory of interferometry. First, some movies to illustrate the problem. Clues of what sort of maths to aim for. Geometrical calculation of phase. Visibility formulas.

2. NASSP Masters 5003F - Computational Astronomy - 2009 Simplifying approximations: I’ll assume a small bandwidth Δ ν, so signals can be approximated by sinusoids. I’ll neglect polarization. I’ll explain interferometry first in 2 dimensions, then extend this to 3. There are many sorts of interferometer but I’ll concentrate on aperture synthesis. The fundamental maths for a 2-antenna interferometer can be easily expanded to cater for more than 2.

3. NASSP Masters 5003F - Computational Astronomy - 2009 Interferometry

4. NASSP Masters 5003F - Computational Astronomy - 2009 Interferometry

5. NASSP Masters 5003F - Computational Astronomy - 2009 Expression for the phase difference φ θ Path difference d = D sin θ D=u λ

6. NASSP Masters 5003F - Computational Astronomy - 2009 Clues..? Since φ is proportional to sin(θ), if we could measure φ, we could work out where the source is in the sky. From last lecture, we saw that correlating the signals from the two antennas gives us a number proportional to S exp(-iφ), where S is the flux density of the source in W Hz -1. Things are looking good! The fly in the ointment is...

7. NASSP Masters 5003F - Computational Astronomy - 2009 The sky is full of sources. The correlation returns an intensity-weighted average of all their phases. D=u λ

8. NASSP Masters 5003F - Computational Astronomy - 2009 Formally speaking... The correlation between the voltage signals from the two antennas is We’ll call V the visibility function. A is the variation in antenna efficiency with θ. I is the quantity formerly known as B – ie the brightness distribution. Its units (in this 2- dimensional model) are W m -2 rn -1 Hz -1 To keep things simple(r) I’ve ignored any summation over frequency ν.

9. NASSP Masters 5003F - Computational Astronomy - 2009 Coordinate systems – everything is aligned with the phase centre. b w u For baselines: 2 1 θ l (1-l 2 ) 1/2 Celestial sphere dθdθ =sin θ The direction normal to the plane of the antennas is called the phase centre. Normally the antennas are pointing that way, too. =cos θ

10. NASSP Masters 5003F - Computational Astronomy - 2009 Putting it all together: gives We want to complete the change of variable from θ to l. It’s not hard to show that so the final expression is

11. NASSP Masters 5003F - Computational Astronomy - 2009 Ahah! Provided w=0, or in other words provided the antennas all lie in a single plane normal to the phase centre, This is a Fourier transform! We just need to get a lot of samples of V at various u, then back- transform to get Trouble is, we can’t always keep w=0.

12. NASSP Masters 5003F - Computational Astronomy - 2009 Non-coplanar arrays. b 1,2 2 1 θ l (1-l 2 ) 1/2 dθdθ 3 b 2,3 b 1,3 Phase centre w u For baselines: u 1,3 w 1,3 = w 2,3

13. NASSP Masters 5003F - Computational Astronomy - 2009 The factor of (1- l 2 ) 1/2 prevents the full expression for V´(u,w) from being a Fourier transform. But for small l, (1- l 2 ) 1/2 is close to 1, and varies only slowly. Let’s do a Taylor expansion of it: The full expression therefore becomes (1- l 2 ) 1/2 = 1 – ½ l 2 +O( l 4 ). Non-coplanar arrays - small-field approximation:

14. NASSP Masters 5003F - Computational Astronomy - 2009 Non-coplanar arrays - small-field approximation: For V(u)=V´(u,w)exp(2 π iw), and πwl 2 <<1, and we are back to the Fourier expression. V is like measuring V´ with a phantom antenna in the same plane as the others. So: for non-coplanar antennas, what matters is the projection length u of each baseline, projected on a plane normal to the ‘phase centre’.

15. NASSP Masters 5003F - Computational Astronomy - 2009 The phase centre The location of the phase centre is controlled as follows: 1.Decide which direction you want to be the phase centre. 2.That direction defines the orientation of the u-w axes. 3.Calculate w j,k for each j,k th baseline in that coordinate system. 4.Multiply each correlation R j,k by the appropriate e 2 π iw factor. (Equivalent is to delay the leading signal by t=w/ ν.) 5.After appropriate scaling, the result is a sample of the visibility function V(u j,k ) appropriate to the chosen phase centre.

16. NASSP Masters 5003F - Computational Astronomy - 2009 Now we go to 3 dimensions and N antennas. The coordinate axes in 3 dimensions are labelled (u,v,w) for baselines and (l,m,n) for source vectors. Each pair of antennas gives us a different projected baseline. N antennas give N(N-1)/2 baselines. Thus N antennas give N(N-1)/2 samples of the ‘coplanar’ visibility function

17. NASSP Masters 5003F - Computational Astronomy - 2009 An example The full visibility function V(u,v) (real part only shown). A familiar pattern of ‘sources’

18. NASSP Masters 5003F - Computational Astronomy - 2009 Let’s observe this with three antennas: u v Longitude Latitude

19. NASSP Masters 5003F - Computational Astronomy - 2009 ‘Snapshot’ sampling of V is poor.

20. NASSP Masters 5003F - Computational Astronomy - 2009 Aperture synthesis via the Earth’s rotation.

21. NASSP Masters 5003F - Computational Astronomy - 2009 View from the phase centre